MATH 152
Exam 2
Fall 1997
Version A
Student (Print)
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Last,
First
Middle
Student (Sign)
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Student ID
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Instructor
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Section
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1-10 |
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11 |
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12 |
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13 |
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14 |
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15 |
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TOTAL |
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Part I is multiple choice. There is no partial credit. You may not use a calculator.
Part II is work out. Show all your work. Partial credit will be given. You may use your
calculator.
Part I: Multiple Choice (5 points each)
There is no partial credit. You may not use a calculator. You have 1 hour.
1. The natural length of a certain spring is at 2 m. A force of 6 N will stretch the spring from
2 m to 4 m. How much work will it take to stretch the spring from 4 m to 6 m?
a. 18 J
b. 24 J
c. 36 J
d. 48 J
e. 60 J
dy
x 2 " y 2 has direction field
dx
2. The differential equation
2
y1
a.
-2
-1
0
1
x
2
1
x
2
1
x
2
-1
-2
2
y1
b.
-2
-1
0
-1
-2
2
y1
c.
-2
-1
0
-1
-2
2
y1
d.
-2
-1
0
1
x
2
1
x
2
-1
-2
2
y1
e.
-2
-1
0
-1
-2
dy
Ÿ2x 1 Ÿy 1 is
dx
neither separable nor linear
separable but not linear
linear but not separable
separable and linear
none of these
3. The differential equation
a.
b.
c.
d.
e.
4. Solve the differential equation
2
dy
3 x 3 with the initial condition yŸ0 2.
4 y
dx
a. x 3/4 2
b. 3 x 4 8
c. x 4/3 2
d. 2x 4/3
e.
5. If a
a.
b.
c.
d.
e.
4
x 3 16
˜4, 2, 1 ™ and b ˜3, 2, 3 ™ then a 2b ˜"2, "2, "5 ™
˜10, 6, 7 ™
˜7, 4, 4 ™
˜1, 0,
"2 ™
˜11, 6, 5 ™
6. The triangle with vertices P Ÿ4, 2, 1 , Q Ÿ3, 3, 1 and R Ÿ3, 2, 3 is
a. equilateral
b. isosceles but not right
c. right but not isosceles
d. isosceles and right
e. none of these
7. In 3-dimensional space the equation x 2 z 2 16 describes
a. the interior of a circle of radius 4 in the xz-plane centered at the origin.
b. the region inside a paraboloid centered along the positive y-axis.
c. the region inside a cone centered along the positive y-axis.
d. the part of the xz-plane outside of a circle of radius 4 centered at the origin.
e. the interior of a cylinder of radius 4 centered on the y-axis.
8. The arclength of the curve y e 2x between x 1 and x 2 is given by
a. ;
b. ;
c. ;
2
1
e4
e2
2
1
e4
1 e 4x dx
1 2e 2x dx
1 4e 4x dx
d. ; 2 2= 1 e 2x dx
e
2
e. ; 2=x 1 e 4x dx
1
9. The curve y sinŸx , between x 1 and x = , is rotated about the x-axis. The
2
surface area of the resulting surface is given by.
=/2
a. ; 2= sinŸx 1 cos 2 Ÿx dx
1
b. ;
c. ;
=/2
1
=/2
1
1
2=x 1 cos 2 Ÿx dx
2= cosŸx 1 sin 2 Ÿx dx
d. ; 2=y cos "2 Ÿy 1 dy
0
e. ;
=/2
1
2=x 1 sin 2 Ÿx dx
10. Find a unit vector in the direction of the vector Ÿ3, "4, 12 a.
b.
c.
d.
e.
12 , " 4 , 3
13 13 13
" 12 , 4 , " 3
13 13 13
" 3 , 4 , " 12
13 13 13
3 , " 4 , 12
13 13 13
" 4 , 12 , 4
13 13 13
Part II: Work Out (10 points each)
Show all your work. Partial credit will be given.
You may use your calculator but only after 1 hour.
11. A plate in the shape of an isosceles triangle with base of 6 m and altitude of 9 m is put
in water with the base at the surface of the water and the point straight down. Find the
total force on the plate. Note the density of water is 1000 kg / m 3 and the acceleration of
gravity is 9.8 m / s 2 .
12. Solve the differential equation
3
dy
3x 2 y 6x 2 e x with the initial condition yŸ0 5.
dx
13. A water trough has the shape of half of a cylinder on its side, as shown in the figure.
The radius is 2 m and the length is 10 m. How much work does it take to pump the
water out of the top of the trough if the trough is full of water? The density of water is
1000 kg / m 3 .
14. Find the x-coordinate of the centroid of the area bounded by the parabola y x 2 and
the lines x 1 and y 0.
15. A barrel initially contains 3 cups of sugar dissolved in 4 gallons of water. You then add
pure water at the rate of 2 gallons per minute while the mixture is draining out of a hole
in the bottom at 2 gallons per minute. Find the amount of sugar in the barrel after 1
minute.
Note: You must explicitly show the differential equation, the initial condition, the method of
solution, the general solution, the solution satisfying the initial condition and the final
answer.